This is a fairly small question - but what happens to what they molt? Do you clean it up? Do they use it for something? I guess I have never thought about it.

The way I like to describe it is like kicking off a blanket. The tarantula will flip over on its back and it will look like slowly kicking off a blanket. When done the tarantula may lay there and rest for a little bit then turn over. When the are done the tarantula will be very fragile (more fragile than usual), its fangs will be white. When the fangs are white they could easily break off. White fangs mean "hey, don't feed me" the tarantula will harden up (you will be able to tell by the fangs), red fangs mean "not yet, I am not quite ready to eat" lastly when the fangs are black the tarantula is ready to eat and is hardened. (Still fragile, as all tarantulas are). It always helps to keep the water dish full and even overflow the dish a little, this will give it extra moisture to ensure a successful molt.